DYNAMICS OF AN OIL CONTAINMENT
BOOM IN A WAVE FIELD
by
Samuel H. Drake
B.S.,
Massachusetts Institute of Technology
(1965)
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June, 1970
Signature of Author.....
...................
.......................
Department of Mechanical Engineering, June 4, 1970
/
/
1, p,
Certified
b ..
..
. ...
,.L
*V*
*
..........................
..........................
Thesis Sup rvisor
Accepted by........
.
................
Chairman, Departmental Committee on Graduate Students
Archi ves
JUL 16 1970
'3f? A RIES
ii
ABSTRACT
DYNAMICS OF AN OIL CONTAINMENT
BOOM IN A WAVE FIELD
by
Samuel H. Drake
Submitted to the Department of Mechanical Engineering on
June 4, 1970 in partial fulfillment of the requirement for
the degree of Master of Science.
A theoretical model was developed for the magnitude and phase
of the surge or horizontal displacement of an elastically constrained flat plate boom model.
An experiment was set up and
data was taken in an attempt to verify the theoretical model.
The
resulting data supported the theoretical model within the range
of the experiment.
Thesis Supervisor:
David P. Hoult
Title:
Associate Professor of Mechanical Engineering
U.
iii
ACKNOWLEDGEMENTS
The author would like to sincerely thank Professor David
Hoult for his guidance and many ideas and Professor Jerome
Milgram for his help with the theoretical background and making
available the use of his precision wavetank.
The author would
also like to thank Miss Sara Rothchild for her help in preparing
the manuscript and his wife, Alice, for her help in proof
reading the manuscript and her acceptance of my irregular hours.
This research was sponsored by the Department of the Interior,
FWPCA contract no. 15080 ESL.
iv
TABLE OF CONTENTS
Title Page
Abstract
Acknowledgements
Table of Contents
List of Tables and Illustrations
List of Symbols
i
ii
iii
iv
v
vii
Introduction
1
Theory
4
Experiment
17
Results
27
Conclusions
30
Appendix A
31
Appendix B
43
References
48
Tables
49
Illustrations
52
V
LIST OF TABLES AND ILLUSTRATIONS
Table 1:
Values of PD' PM, and -tan1 (G/-H) for Ka = 0.05
through 5.0.
Table 2:
Theoretical values for magnitude and phase of X/A
along with values for Q, R, and S for Ka = 0.05
through 2.0.
Table 3:
Experimental values of magnitude and phase of X/A
along with values of wavelength and wave height.
Figure 1:
Schematic of constraint forces on model boom.
Figure 2:
Schematic of constraint forces on full scale boom.
Figure 3:
Close-up photograph of boom in wave tank.
Figure 4:
Overall view of boom in wave tank including elastic
constraint support arms and wave height gauge.
Figure 5:
Sequence of boom positions photographed during run
C5.
Figure 6:
Drawing of measurement grid lines and boom pointers
illustrating measurement points.
Figure 7:
Plot of the theoretical values of the magnitude of
X/A for Ka = 0.05 through 5.0 with T = 0.0, 0.071,
0.142, 0.281.
Figure 8:
Plot of the theoretical values for the phase of X/A
for Ka = 0.05 through 5.0 with T = 0, 0.071, 0.142,
0.281.
Figure 9:
Example of wave height data from run C5 with curves
using first Fourier coefficients and first and second
Fourier coefficients.
m
vi
Figure 10:
Example of surge displacement data from run C5 with
curves using first Fourier coefficients and first and
second coefficients.
Figure 11:
Plot of the theoretical values of the magnitude of
X/A with the experimental data points obtained for
Ka
Figure 12:
=
0.05 through 2.0 and T equal to 0.142.
Plot of the theoretical values of the phase of X/A
with the experimental data points obtained for
Ka = 0.05 through 2.0 and T equal to 0.142.
vii
TABLE OF SYMBOLS
a
boom draft
aX0, aXl, aX2
Fourier cosine coefficients for horizontal displacement
aAO, aAl, aA2
Fourier cosine coefficients for waveheight
A
waveheight
bX1 , bX 2
Fourier sine coefficients for horizontal displacement
bAl, bA 2
Fourier sine coefficients for waveheight
B
damping coefficient
Fc
constraining force
FD
damping force
F1
inertial force
F
added mass force
F
excitation force
g
gravitation acceleration
G
as defined by Eq.
H
as defined by Eq. 18
m
x
17
moment of inertia
nondimensional force coefficient
K
wave number
L
length of tension members
M
mass of boom
M
added mass of fluid
a
P D
damping coefficient
viii
PM
added mass coefficient
Q
as defined by Eq. 52
R
as defined by Eq. 53
S
as defined by Eq. 51
T
tension force
V
magnitude of boom velocity
x
horizontal distance
as defined by Eq. 15
as defined by Eq. 16
reflected wave potential coefficient
wave surface elevation
ii
phase angle between surge and waveheight
CA
X
X
0
phase angle of waveheight data
phase angle of surge data
phase angle between exciting force and surface
elevation
X
wavelength
displacement from centerline
p
density of fluid
T
nondimensional tension coefficient
reflected wave potential
X
horizontal displacement coefficient
phase angle due to y±
W
frequency of waves in radians/sec.
- 1 INTRODUCTION
In recent years there has been a large increase in the amount
of oil both transported across the oceans in tanks and produced
on offshore
platforms.
Along with this increase has come the seem-
igly inevitable increase in the amount of oil spilled on the oceans
and bays with the attendant destruction of wildlife and the contamination of beaches.
Some of these oil spills such as those caused
by the practice of pumping contaminated ballast water overboard, can
be easily prevented.
While greater care might prevent some of the
accidents, particularly those occuring during loading and unloading,
it is highly probable that oil spills will continue to be a serious
pollution problem.
In the past,
three approaches have been taken to control oil
slicks; chemical dispersants or emulsifying agents, burning, and
mechanical containment.
While the detergents and emulsifying agents
may cause a disappearance of the physical oil slick,
the oil is not
removed from the oceans but rather it is only dispersed and mixed
with the water or allowed to sink to the bottom.
be removed by natural decomposition.
The oil must still
Also some of the chemicals
that have been used have also proved to be harmful to the natural
environment.
Burning or partially burning off the oil slick is some-
times possible particularly if the slick is composed of the more
volatile compounds and if the slick is not too thin.
The primary
problem with burning off the slick is that the heat loss thru the
slick to the water is so great that the temperatures needed to sustain the combustion cannot be maintained.
Recently materials have
- 2 -
been introduced that are designed to act as a wick for the oil and
thus draw some of the oil up away from the cold water by capillary
action permitting the necessary temperatures for combustion to be
sustained.
Even in this case it is unlikely that all of the
heavier or less volatile materials will be burned.
Also, there is
a large amount of oily smoke produced which would be objectionable
near populated coastal areas.
The best alternative might be mechanical containment and subsequent collection if it were possible, as not only would the oil
be totally removed from the ocean but the economic value of oil
could also be recovered.
In the past, a number of different de-
signs of both solid (although flexible) and pneumatic or inflatable
oil containment booms have been tried ranging from a simple line
of logs chained together to a structure made by lashing a long
string of barges together.
The use of air bubble curtains has
also been considered although the power needed appears to be too
great for any large scale use.
Some of the mechanical booms have
worked quite well inside harbors or other protected waters, but thus
far the booms have not been satisfactory when higher winds, waves,
or currents are present.
The purpose of this thesis was to study the dynamics or motion
of an oil containment boom model under the influence of waves.
A
theoretical model is presented which predicts the magnitude of the
ratio of the surge or horizontal displacement to thewave height and
the phase lag between the surge displacement and the uave height for
I*
- 3 an elastically constrained flat plate boom model aligned parallel
to the incident waves.
This theoretical model considers the
elastic constraining force, the inertial force, the exciting force
due to the incident wave and both a drag and added inertia force
due to the reflected wave.
As the exciting force due to the in-
cident wave and the forces due to the reflected wave are expressed
in terms of a damping coefficient and an added mass coefficient
which are a functions of the nondimensional term Ka where K is the
wave number and a is the boom draft, the nondimensional ratio of
the surge displacement to the waveheight is presented as a function
of Ka.
Experimental data on this motionwere taken with a suitably
scaled flat plate boom model suspended in a wave tank varying the
parameter Ka over an order of magnitude range in the area of interest and varying the waveheight to the point of breaking waves.
This data was then used in an attempt to experimentally justify
the theoretical model.
This theoretical model could then be used
to predict the motion of a boom with a more complicated shape if
the damping and added inertia coefficients can be experimentally
determined as a function of Ka.
This coupled with a knowledge of
the boom behavior under conditions of wind and current should permit the designing of oil containment barriers to operate satisfactorily under open sea conditions.
I
- 4 THEORY
As the theoretical values for the damping and inertial coefficients for a swaying or surging (the horizontal motion of a boom
being referred to as surge and the vertical motion being referred
to as heave) flat plate are presently available, the model was
limited to a flat plate.
The forces on the boom considered by the
theory were limited to the constraining forces, the inertial force,
the force due to the incident wave and the forces due to the reflected wave.
surge.
Only one degree of freedom was considered, that of
However, a similar theory would hold for heave and roll.
The boom or flat plate was constrained by an elastic member
at each end such that the forces were purely tensional at zero
displacement (see Figure 1).
If x is the horizontal or surge
displacement, the constraining force in the x direction is
-x L(x)
where T is the tension force applied by the elastic
members and L is the length of the elastic members.
If L >> x,
then the variation in length and the change in T are both small
and both L and T may be considered as constants or
F
c
=2Tx
L
In order to determine the appropriate tension force, a nondimensional force coefficient, J, was defined as the ratio of the constraining force per unit length of boom to the exciting force of
the incident wave.
As will be shown the exciting force per unit
length of boom is proportional to pgAa where p is the water density,
- 5 g is the gravitational acceleration,
is the boom draft.
A is the waveheight and a
Therefore the nondimensional term, J, for the
two dimensional flat plate model parallel to the incident waves is
=
(2)
2Tx
Lk pgAa(2
where Z is the length of the boom.
In the full scale case, the boom cannot be considered as a
rigid body and the constraining force per unit length must be
found in terms of a differential displacement.
It is assumed
that the displacement from the relaxed position is proportional
to the waveheight and is sinusoidal with the same wavelength as the
incident waves or
A sin ( 2rx
where
(3)
is the displacement from the relaxed position and X is the
wavelength of the incident wave.
Considering a small element
Ax
of boom, the components of the tension force that are acting to
constrain boom to its relaxed position are
|1 and
a1
|2
where the
ax 2
subscripts 1 and 2 refer to the left and right end of the boom
Therefore the constraining force per unit
element (see Figure 2).
length on this element of boom is
(1
F
c
=T'
- KI)
2
Ax
1
- 6 where the use of the prime mark indicates the full scale conditions.
At the limitAx+ 0
2
F
c
=
T' 3
(5)
2
or using Eq. 3
F
c
T'A' (2)2
(6)
,2
Remembering that the displacement was proportional to the waveheight:
2
F '
c
~T'x'(27)
(7)
,2
Therefore the ratio of the constraining force to the exciting
force for the full scale case is
,
T'x' (2r)
X,
2
2
(8)
pgA'a'
Setting J and J' equal, the value T/L for the model can be found
in terms of T' and
2Tx
Lk pgAa
'
T'x'(27)
X
2
2
pgA'a'
(9)
- 7 T'(2r)2a
2X 2 a'
T
L
(10)
where it was assumed that the ratio x'/A' was equal to x/A.
While the force coefficient J is nondimensional, it is a
function of the nondimensional ratio x/A, the horizontal displacement divided by the waveheight.
As will be shown, the term x/A
is in turn a function of the nondimensional term Ka.
A somewhat
more convenient relationship that expresses the tension force in
nondimensional terms and avoids this problem is
(11)
T = -
x/A
or making use of Eq. 2
2T
T=
LUpga
(12)
The inertial force is simply
2
F
=-
md x
dt
(13)
where m is the mass of the boom.
If the boom is assumed to surge back and forth with a sinusoidal
motion (if the motion is cyclic but not sinusoidal it can be broken
down into Fourier components), the motion can be represented as
x = X cos wt or x = Xe
iwt .
erated wave elevation as
For this motion Ursell
1
gives the gen-
- 8 Tl(x,t) = X
ra I 11 (Ka)
(14)
+ L1 (Ka)I Re [eikx + iwt
-itanl (G/-H)
where
=
1/7
[ 1 (Ka) + K (Ka)] 1/2
(16)
- Kab I(Ka) + L (Ka)
=
G =
(15)
2 aI
(Ka)a 2 /Ka
(17)
H = K1 (Ka)G/nI1 (Ka) = nra 2 K (Ka)/Ka
and Il, K1 , L 1are functions defined by Watson
(18)
2
.K
is the wave
number
K
(19)
1/X = w /g
and a is the depth or draft of the flat plate in the undisturbed
fluid.
Kotik3 defines the dimensionless damping and added inertia
coefficients PD(Ka) and PM(Ka) respectively where
F
PD(Ka) =
D__
2
MaWX
a
and PM(Ka)
-
FM
2
Ma WX
a
(20)
- 9 F
D
and F
M
are the amplitudes of the force per unit length exerted
by the plate on the fluid in phase with the velocity and the acceleration of the plate and Ma is the mass of a semicircle of fluid
of radius a
(21)
Ma = pfra /2
The coefficients are given as:
2
27rx
PD(Ka) =
(22)
2 111 (Ka) + L1 (Ka)I
(Ka)
P (z) dz
M
M
z-
Tf
Ka
0
Values for PDand PM along with values for -tan
(G/-H) are given
by Kotik as a function of Ka in tabular form (see Table 1).
Therefore,
if
the boom motion is given by x = X eit,
the
damping force exerted on the boom is
FD =
X ei
PD(Ka)MaW
t --P (Ka)Maw
(/2
Xeist + i
and the added inertia force on the boom is
FM =P M(Ka)Maw
X eiot = -P
(Ka)Maw2
X eist + i
( 25)
Knowing the velocity potential for waves caused by a forced
oscillation of the plate in calm water,
it
is possible to determine
the exciting force on the plate due to incident waves by using
-
Haskind's4 relations.
10
-
If the velocity of the plate during the
forced oscillation is represented as V eiot, the asymptotic radiated
potential for x
o is given as
-
(26)
<p = V Y± eKz - iKx + iwt
where the function y
is only a function of the wave number K and
the boom geometry and the superscript ± refers to the case at
x +±
0.
For the case of a symmetric body y
= - y
.
From Eq. 26
it follows that the far field surface elevation is
n(x,t) = V y
-ikx
it
1
(27)
+ it
This can also be written as
n(x,t) = V1y I T
where
IP
eikx +
t + $
is the phase angle due to Y~.
(28)
Integrating the expression
for the velocity of the plate to obtain the displacement,
x =
it
e
(29)
is seen that 4) is the phase angle by which the surface eleva-
tion at x + ± 0 leads the body displacement.
If the velocity potential of an incident wave is considered
to be
-
. E
0
11 -
kz - ikx + iwt
(30)
W
where A is the wave amplitude, the exciting force per unit length
for a two dimensional body is given by Newman5 as
F
= pg A y- e-iWt
(31)
Using the same notation as before where $
to y
is the phase shift due
the exciting force can be written as
F
x
= pgA
Iy I
eiWt + i*
(32)
From the velocity potential of the incident wave the surface elevation is seen to be
n0 (x,t)
A
-ikx + iwt M - iA e-ikx + iwt
(33)
or
0 (x,t) = A e-ikx + iwt -
(34)
i7/2
Comparing this result with the equation for the exciting force, it
can be seen that the phase angle between the exciting force and the
surface elevation is $~ + 7/2.
As y
= -Y , $
=
$,
-
7.
If
e is
defined as the phase angle between exciting force and the surface
elevation,
-
6= $~ + -n/2 =
t+
A damping coefficient,
12
-
(35)
-n/2
which is the force in phase with the
velocity of the oscillating plate divided by the velocity, can be
determined as a function of y
in the radiated waves.
by considering the energy carried
For a symmetrical, two dimensional body,
this is given by Newman as
2
B = p(y-)
(36)
Therefore the damping force per unit length is
iWt=PWIY
- B V e
FD
2Veit
(37)
From Eq. 24 this damping force is seen to be
F
D
or as V e
= IP (Ka) MW
D
a
it
(24)
= iWX et
B = PD(Ka)Ma w
(38)
or using Eq. 36
plyI
2
=
PD(Ka)Ma w
(39)
(1/2
[PD(Ka)Ma
p(40)
- 13 -
2
as M
=
a
p2Ta
2
I~I
[P D (Ka) 'ra
2
=
(41)
1/2
As the surface elevation for the radiated waves was given in Eq.
14 as
1
rarI1 (Ka) + L1 (Ka)IRe[eikx = iwt - i tan- (G/-H)]
fl(xt) = X
the phase angle between the surface elevations at x
motion is - tan
-
(14)
co and the body
This is exactly the same angle as $+, the
(G/-H).
phase angle shift due to y+
Therefore the exciting force per unit length due to an incident
wave with a surface elevation
n(x,t) = A e-ikx + iwt
(42)
is
2
F
x = pgA
=pa APD(Ka) iwt + ie
e
2
2
(43)
where e is the phase angle of the force with respect to the surface elevation of the incident wave
aI+
-
= - tan~ 1 (G/-1)
-2(4
-tn
G-H
-
(44)
- 14 Assuming that the forces due to the incident waves and the
radiated waves are linearily additive and that the incident wave
is of the form A e
t and the boom displacement is X eiWt where
A is the wave height and X is a complex expression containing both
amplitude and phase information, the equation of motion can be
written as
2
F = F
2
FC
+ FD + FM+
(45)
dt2
C
x D 14
or substituting the expressions derived for FX, FC, FM, FC and F :
Pg A at
a
st+
02
- P (Ka)M w
PD(Ka) et+i
a
D
2 D
X e
iwt + inr/2
(46)
-P1 4(Ka)Maj
a
M
2
X e
iwt + ir
2T
eit
-- L7X e
-MLO
2
Xe
iWt + iT
where the forces that were.previously expressed as forces per unit
length are multiplied by the length of the boom,
due to the motion of the boom,
F
and FM,
£.
For the forces
this is accomplished by
considering the apparent mass term Ma to be
Ma
I22 a2
Dividing all the terms of the equation of motion by e
bining terms and simplifying
(47)
, com-
I
- 15 -
pgAa
-i7
K+
X[(M +
P D(Ka) e
)2T
[M+P m(Ka)M a)W
T
(48)
- i PD(Ka)M aW2 ]
=
0
or
-pg
A
at
2P (Ka) e
2 D
(M + P m(Ka)M a )
Ti
(49)
D(Ka) Ma
If this is written in the form
X
A
where
Se 10
(50)
Q + iR
S = -pg at
j
(51)
PD(Ka)
2T
22
Q = [(M + P (Ka)Ma)W
m
--
a
L
(52)
]
R = - PD(Ka)Ma 2
(53)
can be expressed as
-
A
X
A
or
i tan (-R/Q)
Q2 + R2)1/2
2
Q +R
2
ie
(54)
-
X
A
_Se10
+ i tan
16
-
(-R/Q)
(Q2 + R )1/2
As S, Q, R and 0 are only a function of Ka, the magnitude and phase
of X are therefore a function of the nondimensional term Ka which
is the desired result.
With this information it is possible to
predict the motion of a flat plate or any other shape if the functions PD(Ka) and PM(Ka) are known or experimentally determined.
-
17 -
EXPERIMENT
The experimental part of this thesis consisted of taking data
on the motion of an elastically constrained flat plate boom model
suspended in a wave tank.
The basic design constraints were that
the apparatus be designed to make use of the precision wave tank
of the Naval Architecture and Marine Engineering Department and
that the data would be first taken on film with 16 mm motion picture equipment and then transferred onto computer cards.
The precision wave tank is approximately three meters long
with a cross sectional width and height of 30 cm and 20 cm respectively.
The mean operating water depth used during the experi-
mental runs was 15 cm.
The waves are generated with a sinusoidal-
ly driven paddle that pivots from the bottom of the tank.
Both
the frequency and the amplitude of the oscillation are variable,
the practical amplitude being limited to the case of breaking
waves and the frequency range being limited by both the finite
depth effects and the reflection coefficient of the wave absorbing
beach for the long wave lengths and the rate at which the paddle
could be driven for short wavelengths.
The scaling ratio between
the experimental model and the full scale conditions is largely
determined by the acceptable range of wave lengths that can be
generated in the wave tank.
This range was from approximately 10 cm to 100 cm with the
longer wavelengths well inside the area where the wavelength as a
function of the frequency must be corrected for the effect of
- m
- 18 The relationship of frequency to wavelength for
finite depth.
finite depth is
V2 = --- tanh (27Th
(56)
or
2
2 =
g~r
tanh (
2Th
)
(57)
where V is expressed in cycles per second and w in radians per
second.
At a wavelength of 100 cm and a depth of 15 cm the finite
27Th
h) is approximately equal to 0.73
depth correction term tan H(- -and does not closely approach one until a wavelength of 30 cm
(tan H(
)
=
0.995).
If the wavelength region of interest for full scale waves is
approximately 3 to 30 meters and the full scale boom height or
draft is approximately one meter then, keeping the nondimensional
ratio of the wavelength to boom height constant, the height of
the boom model should be approximately 3 cm.
Assuming an effective
specific gravity of the boom of 0.8, then 2.4 cm of the boom height
will ride below the calm water level or a = 2.4 cm and the range of
Ka for the experiment-is approximately 0.15 to 1.5.
The constraining force for the boom model is found by using
the nondimensional ratio of the constraining force to the exciting
force, J, as expressed in equations 2 and 8
-
-
19 -
2Tx
pgAa
L
(2)
T'x' (27)
2
(8)
,2 pg'a'
where, as before, the prime marks are used to indicate the full
scale conditions.
Equating J and J' yields the relationship
given in equation (10)
2
T = IT (27r)
a
L
2
2X, a'
(10)
If typical full scale conditions are assumed to be a tension force
of 5000 lb or approximately 2.5 x 109 dynes (this condition is found
by considering a full scale boom to be anchored at the ends and in
a current), a wavelength of 30 meters and a ratio of a/a' as 1/30,
T/L should be approximately 5 x 103 dynes.
The length of the
elastic constraining members is somewhat arbitrary, the main limitation being that the length be much greater than boom displacement
so that the constraining force remains linear with the displacement.
With the expected boom displacement not to exceed 5 cm., the length
L was set at 50 cm.
5
set at 2.50 x 10
The tension force for the model was therefore
dynes.
This tension force was divided by two and
effectively applied at one centimeter above and below the centerline
of the boom in order to provide a roll constraint or restoring moment.
This two centimeter separation was somewhat arbitrary although
L
-4
-
20 -
representative of the full scale booms which have a cable at top and
bottom.
The separation distance could have been modified had it been
necessary to change the roll resonance frequency.
It was originally considered desirable to correctly model
the moment of inertia of the boom.
The nondimensional parameter
that governs this is the ratio of the moment of inertia per unit
length of the boom about its roll axis to the moment of inertia
per unit length of a semicircular cylinder of fluid of radius
around its axis.
(58)
4I
npa /4
where (D is the nondimensional parameter and I is the moment of
inertia per unit length to the boom.
If the full scale boom is
considered to be rectangular in cross section and of uniform density then the moment of inertia per unit length is
M'h'2
1'
(59)
4
where M' is the mass per unit length and h' is the height of the
boom.
Therefore
e,
.
M'2
'h'2
(60)
ipa' 4
Setting D equal to
'
- 21 -
I
4
TPa /4
M'h' 2
7rpa
(61)
,4
I= M'h' 2 a
(62)
4 a'4
Taking as typical full scale conditions a mass per unit length,
M', of 1.6 x 103 gm/cm, a boom height, h', of 100 cm, and a boom
draft of 80 cm, the moment of inertia per unit length of a model
boom with a draft of 2.4 cm should be 3.24 gm cm.
Multiplying
this by the length of the boom, 30 cm, the total moment of inertia
should be approximately 100 gm cm2
The final boom model was a flat plate 30 centimeters long,
3 centimeters wide and 0.7 centimeters thick.
The thickness is
determined by the buoyancy necessary to float the boom and the
supporting members such that 20% of the boom is out of the water in
still conditions.
In order to transmit the constraining force
over the side walls of the tank and allow the boom 5 centimeters
of vertical motion an elongated horseshoe shaped piece was added to
each end of the boom.
To shift the center of mass of the assembly
below the center of buoyancy such that the boom was stable in an
upright position, a counterweight mass was added to a piece of
thin aluminum tubing projecting down from the outside arm of the
support member on either end of the boom.
(see figure 3).
The
boom was made of laminated balsa wood with the end support members
made of a lamination of balsa wood and adhesive backed aluminum
I
-22
foil.
-
The boom and support members were painted with a thin epoxy
base paint to waterproof the balsa wood.
With the brass counter-
weights added, the total mass was 39 grams.
The amount of inertia of the .actual boom model was measured
by constructing a torsional pendulum using a length of fine piano
wire.
The torsional spring constant was calculated by measuring
the period of oscillation with a body of known moment of inertia.
The measured moment of inertia for the final boom model was ap-
2
proximately 2200 gm cm
or approximately 22 times too large.
This
large moment of inertia is largely due to the mass of the end support members and their counterweights.
The effect of the large moment of inertia was not deemed important, however, as the roll resonance occurred at a wavelength
of approximately 150 cm or outside the region of interest and the
surge and roll were not strongly coupled in the region of interest.
An experimental test was run at a wavelength of 66 cm using two
boom models, one with a moment of inertia of 5200 gm cm
other with a moment of inertia of 2200 gm cm
2
.
2
and the
Similar results
were obtained in both cases.
On one end of the boom the tension forces were applied by a
pair of thin rubber strips which were held clamped 50 centimeters
from the end of the boom.
On the other end of the boom the ten-
sion forces were applied by a pair of lightweight nylon cords which
were also clamped 50 centimeters from the end of the boom.
The
exact length of the nylon cords was adjustable with screws such
that the boom could be centered in the wave channel.
The tension
-
23 -
is adjusted by running the rubber strips over a ball bearing
pulley and attaching a weight pan and laboratory weights and then
clamping the strips when the correct tension is obtained.
A 25 centimeter long section of one wall of the wave tank was
made of transparent plastic so that the motion of boom and the water
could be photographed from the side perpendicular to the travel of
the waves.
In order to reference the position of the boom and the
wave height and to provide a reference length the plastic plate was
scribed with a set of lines (see figure 6).
A set of pointers
was attached to one of the support members of the boom so as to
mark the relative position of the boom.
The boom motion was photographed using a Hycam high speed 16 mm
movie camera run at approximately 100 frames per second.
The pri-
mary reason for using the high speed camera and the relatively fast
framing rate was to obtain a shutter speed fast enough such that the
motion of the boom would not be blurred.
On motion picture cameras
the shutter speed is a function of the framing rate.
As the cam-
era needed approximately 10 seconds to achieve a regulated speed,
the camera was run approximately 12 seconds for each data run with
the last few seconds of the film being used for data.
Prior to
making a sequence of data runs a calibration run was made to record
the exact boom position and watr height under still conditions.
After processing the film, positional data was read off the
individual frames using a machine that projects an image of the
frame on a ground glass screen with an x and y cursor.
By moving
the cursors to the point to be read and pressing a button the
- 24 information is punched directly on computer cards in a form suitable
for data processing.
In this way the x and y postion of a reference
point are marked on, along with the y position of the water surface
along the left reference line, the x and y position of the top boom
pointer and the x position of the bottom pointer.
By marking on a
reference point on each frame, it is not necessary to precisely align
each of them.
The number of frames of film read for each data run
depended upon the wave frequency with enough frames being read for
at least 1 1/2 complete wave cycles.
The format for the calibration
frames was the same with the addition of the x position of the
right reference position in order to calculate a scaling factor.
Five frames of data were read for each calibration point.
The frequency of the water waves was measured using a capacitance probe and a paper chart recorder.
The probe consisted of an
insulated wire sticking down into the water with the water acting as
the outer conductor.
The change in capacitance caused by the change
in water height as the wave swept past caused a change in the frequency of an oscillating L-C circuit which is beating with a reference circuit.
The beat frequency provided a measurement of the amp-
litude of the wave.
As the information from this probe was only
used to provide a measurement of the frequency or the wavelength
using Eq. 57, it was not necessary to synchronize this information
with the data taken on film.
The data was analyzed using a Fortran program which calculated
the horizontal and vertical displacement of the center of mass of
the boom plus the roll displacement and the waveheight for each
- 25
data frame.
-
(See Appendix A for a list of the equations used).
After computing the number of frames in one wave cycle, a Fourier
analysis was performed on the wave height and the horizontal and
vertical displacement using the IBM system 360 scientific subroutine for the Fourier analysis of a tabulated function.
The wave-
height and the horizontal or surge displacements were each plotted
as the tabulated function (actual data) and as the function generated by using the first Fourier coefficients (i.e., x(t) - a
0
+ a 1 cos wt + b1 sin wt) and by using the first and second coefficients (i.e., x(t) = a0 + a1
cos wt + b 1 sin wt + a2 cos 2wt
+ b 2 sin 2wt) using the Calcomp plotter (see figures 9 and 10).
The ratio of magnitude of the horizontal displacement to the waveheight was found by using the square root of the sum of the squares
of first
Fourier coefficients.
2
2
aX1
+ bX1
(63)
aA2 + bAAl
The phase angle of the waveheight and the horizontal displacement
was taken to be the arctangent of the first Fourier sine coefficient
Fourier cosine coefficient or
divided by the first
ca
= tan
C
= tan
1
(bAl/aAl)
(64)
(b X1/aXl)
(65)
-
26
-
Therefore the phase lag between the horizontal displacement and
the waveheight is
C= A
-
C
(66)
Data was taken for values of Ka of approximately 0.15, 0.20,
0.30, 0.40, 0.50, 0.75 and 1.40.
At each value of Ka except
Ka = 1.40 several runs were taken varying the amplitude of the
waveheight up to the case of nearly breaking waves.
only one run
At Ka = 1.40
was possible as the wave would break with the ampli-
tude of the paddle drive set to any position other than the
lowest.
1i
- 27
-
III
RESULTS
Theoretical values for the magnitude and phase of X/A were computed for Ka from 0.05 to 5.0 making use of Eq. 55
1X
X=.Se
S+
itan 1(-R/Q)
(55)
2 + R2)1/2
A
where Q, R, and S are defined by Eq. 52, 53, and 51.
(See Appendix
These values were computed for
B for the computation scheme.)
T = 0, T = 0.071, T = 0.142, and T
=
0.281 where T is the nondimen-
sional tension parameter:
2T
Lk 7rga
(12)
The results are expressed in tabular form in Table 2 and graphically
in figures 7 and 8.
From figure 7 it is seen that the parameter
has little effect on the magnitude of X/A for Ka greater than 1 and
that the effect is not strongly pronounced until Ka is less than 0.4.
The reason for this can be seen looking at Eq. 49
X
A
(
PD (Ka) e
2-pga
[M+P (Ka)M I
M
a
2
2T_
L
(Ka)MW 2
D
a
Both the damping force term and the inertial force term are a function of w2 while the constraining force term is constant.
Thus for
small values of Ka where w is small, the constraining force term is
the dominant term in the denominator of Eq. 49.
The exciting force
-
28
-
is not directly dependent on w either, although the value of PD is
a function of Ka.
Thus the damping forces and inertial forces domi-
nate at large values of Ka and the ratio X/A becomes small.
Varying the value of T causes a small shift in the phase of
X/A with the magnitude of the shift declining with increasing Ka
as can be seen from figure 8.
curves that occur
The small oscillations in the phase
for small values of Ka with T not equal to zero
are caused by Q being less than zero when the -
2T
term dominates
Q = [M + P M. (Ka)Ma 12 - 2T
L
(52)
A total of 16 different experimental runs were made varying the
The
ratio Ka by changing the wavelength and varying the waveheight.
results of these runs are displayed in tabular form in Table 3 and
graphically in figures 11 and 12.
The letter prefix to the run num-
ber is just for identification and has no other significance.
The
number simply refers to the pin position used on the apparatus that
generates the wave.
The waveheight is not a simple function of this
number as it also depends on the wave frequency.
The smallest value
of Ka used was approximately 0.15 and the largest value available
was approximately 1.4.
The experimental results seem to fit the theoretical curve for
the magnitude of X/A quite well (see figure 11) although there is
some scatter.
It would have been desirable to have taken data nearer
the predicted peak and on the other side of the peak if this had
been possible.
However, this would have meant operating with waves
- 29 several meters long in a tank only 15 cm deep.
Also, it would have
been better to have taken a few more points in the range of Ka from
0.8 to 1.4 although the range of Ka from 0.15 to 0.5 was considered
more interesting from the point of checking the theory.
The results
might have been better in regard to scatter if either a larger
film format had been used or if a different focal length lens had
been used with small amplitude waves such that the range of motion
came closer to filling the entire film frame (owing more to the
nature of the machine used to read the film than to the actual film
resolution).
The experimental results for the phase of X/A are certainly of
the right order of magnitude, with all except one point being within 0.2 of a radian (-12*) of the theoretical value, but do not show
any clear trend as the theoretical results do not have much variance
in the range of Ka from 0.15 to 1.5 (see figure 12).
In order to
perform a better experimental test of the phase data, it would be
necessary to extend the range of the experiment out to Ka equal to
5 or more to see whether the phase shifts from approximately -7r/2 to
- ir.
This was not possible, however, with the existing apparatus
as the paddle was being driven at the maximum frequency to obtain a
value of Ka equal to 1.4.
Again, better results with regard to scat-
ter could have been obtained with better filming.
III
wkA9
q
- 30 CONCLUSIONS
The response of the boom to a sinusoidal exciting force was
found to be sinusoidal.
Even for the worst case with nearly
breaking waves when the magnitude of the second harmonic of the
wave amplitude was 25 percent of the magnitude of the first harmonic of the wave amplitude, the magnitude of the second harmonic
of the response was less than 10 percent of the first harmonic of
the response (the accuracy of measurement was about the same order
of magnitude).
As the motion was sinusoidal,
the problem is analogous to a
driven harmonic oscillator with a damping term and an added mass
term that are a function of the driving frequency.
As can be seen
from the data, the response is proportional to the wave height
amplitude for constant values of Ka.
Therefore a linearized theory
can be used to describe the magnitude and phase of the response of
the boom to the wave-induced forces.
Within experimental accuracy,
the data supported the detailed theoretical model presented for the
magnitude and phase of the ratio of the surge of the boom to the
wave height of the incident waves.
III
-31APPENDIX A
For each frame of data, the x and y position of a reference
point, the wave height along a reference line, the x and y position
of the top boom marker and the x position of the bottom boom marker
The x position of a second reference point was also
were read.
read for the calibration frames.
These seven points and the grid
lines are illustrated in Figure 6.
The first step in the data
reduction is the determination of a scaling factor KCAL.
KCAL =
(A-1)
XB
XCAL2 - XCAL1
where XCALl and XCAL2 are the measured x positions of the reference
lines and XB is the known distance between them.
The boom position
and water height for static conditions are then found.
The water
height, HCAL, is given by
(A-2)
HCAL = KCAL (YH - YCAL)
where YH is the measured reference value and YCAL is the water
height along the reference line.
The boom roll angle SIGCAL is
given by
SIGCAL = SIN
1
(X2 - X 1
(A-3)
where X1 and X2 are the measured x positions of top and bottom
boom markers.
is given by
The static x positions of the top boom marker, X1CAL,
- 32 (A-4)
X1CAL = KCAL (Xl - XCALl) - XA
where the factor XA is used to transfer the x reference position
to the center grid line.
The static x position of the center of
is given by
the boom, XGCAL,
XGCAL = X1CAL + A sin (SIGCAL)
-
2 cos (SIGCAL)
(A-5)
where A is the distance from the tip of the pointer to the center
line of the boom.
The static y position of the boom is found in
a similar manner.
After determining the position of the boom and the waterline
for static conditions, the surge, heave and roll displacement of
the boom and the ware height were found for the dynamic conditions
by using the same equations and subtracting the static or calibraThis data was then plotted as a function of the
tion values.
frame number if so desired.
The next step was the calculation of the wavelength from the
frequency using a recursive technique to correct for the finite
depth effect.
The initial approximation, XLAM1 , the infinite
depth value was
XLAM
(A-6)
2
G
=
27T(FREQ)
The correction factor,
CORR,
for finite depth was
'III
- 33 -
CORR = tanh 2iDEP
(A-7)
XLAM
where XLAMswas the current wavelength approximation where DEP was the
depth of the water.
ERR
=
From these an err function, ERR, was generated
- XLAM
CORR
(A-8)
1
The next approximation to the wavelength was taken to be
XLAM = XLAM - 0.7 x CORR x XLAM
(A-9)
This was repeated until the err function was less than the desired
limit.
Prior to calculating the Fburier coefficients it was necessary
to calculate the number of data points per cycle.
This was done by
counting the number of data points from the place where the waveheight first
changed the sign or went thru the origin until it
changed sign for the third time.
The first and second Fourier coef-
ficients were then found for the waveheight and the surge and heave
displacements by making use of a subroutine from the IBM scientific
subroutine package.
A function was generated by using the first coef-
ficient and by using the first and second coefficients.
These two
functions were plotted along with the actual data for comparison.
The final step was to calculate the magnitude and phase of
X/A.
The magnitude of X/A was found by dividing the square root of
the sum of the squares of the first Fourier coefficients of the
III
-
34
-
surge displacement by the square root of the sum of the squares
of the first
Fourier coefficients of the wave height.
2
2
ACOFXG(2)
+ BCOFXG(2)
ACOFH(2)
(A-10)
2
2
A
+ BCOFH(2)
where ACOFXG(2) and BCOFXG(2) are the first fourier coefficients
of the surge displacement and ACOFB(2) and BCOFH(2) are the first
Fourier coefficients of the wave height.
The phase of X/A was
found by taking the difference between the phase of surge displacement and the phase of the wave height after the phase of the waveheight was shifted an amount
2ir(XA)
to correct for the fact that
the wve height data was taken along the left grid line instead of
the centerline
(XA is the distance between these two lines).
The
phase of each was found by taking the arc tangent of the first
Fourier sine coefficient divided by the first
ficient.
ourier cosine coef-
For example
XGPHAS = tan
-l
BCOFXG(2)
ACOFXG(2)
where XGPHAS is the phase of the surge displacement.
A listing of
the program follows.
III
DIMENSION H(300),
SIG(300),
XG(300),
YG(300), Z(300) , YO(300),
$HR(300),
FRBRD(300),
ACOFH(5),
BCOFH(5), ACOF XG (5)
BCCFXG(5),
$ACOFYG(5),
BCOFYG(5),
XGFOR(300),
XGVEL(300),
XGACC 3CC),
$YGFOR(300), YGVEL(300) , YGACC(300)
C
C
INITIAL SETUP
C
NOP = 0
G = 980.35
PI = 3.14159
DEP = 15.0
YAC = 10.0
YA = 10.0
D = 2.40
XA = 5.0
XB = 10.0
A = 1.30
B = 3.0
KCALA = 0.0
N = 0
HCALA = 0.0
SIGCA = 0.0
X1CALA = 0.0
YLCALA = 0.0
READ(5,1) NMON,
1 FORMAT (514)
U'
NDAY,
NYEAR,
NRUN,
KAL
C
C
CALCULATE
CALIBRATION
FACTORS
C
00 3 1 = 1, KAL
READ(5,2) XCAL1, YCAL, YH, X1, Y,
2 FORMAT (7F10.9)
XCAL = XCAL2 - XCAL1
KCAL = XB / XCAL
KCALA = KCALA + KCAL
HCAL = KCAL * (YH - YCAL) - YAC
X2,
XCAL2
HCALA = HCALA + HCAL
SIGC = ARSIN (KCAL * ((X2 - Xl) / B))
SIGCA = SIGCA + SIGC
X1CAL = KCAL * (XI - XCALl) - XA
X1CALA = X1CALA + X1CAL
YICAL = KCAL * (Yl - YCAL) - YAC
Y1CALA = YICALA + YiCAL
N = N + 1
3 CONTINUE
KCAL = KCALA / N
HCAL = HCALA / N
SIGCAL = SIGCA / N
X1CAV = X1CALA / N
X1CAV + A * COS(SIGCAL) + (B / 2) * SIN(SIGCAL)
XGCAL
YlCAV = YlCALA / N
YlCAV + A * SIN(SIGCAL) - (B / 2) * CCS(SIGCAL)
YGCAL
WRITE(6,4)
4 FORMAT ('ICALIBRATION DATA ')
WRITE (6,5)
YGCAL')
XGCAL
SIGCAL
HCAL
KCAL
5 FORMAT ('0
SIGCAL, XGCAL, YGCAL
WRITE(6,6) KCAL, HCAL,
6 FORMAT (5F10.4)
C
C
SETUP
FOR DATA
RUNS
C
00 25 L = 1, NRUN
READ(5,7) MRUN, M, FREQ
F10.4)
7 FORMAT (214,
WRITE(6,8) MRUN, NMON, NCAY, NYEAR
8 FORMAT ('1DATA RUN',15,'
DATE:',13,'/',12,'/',12)
WRITE(6,9)
YG')
SIG
XG
H
9 FORMAT ('0 N
C
C
C
DATA REDUCTION
DO
12
I =
1, M
0%
READ(5,10) XCAL, YCAL, YH, Xl, Ylv, X2
10 FORMAT (6FI0.9)
H(I) = KCAL * (YH - YCAL) - HCAL - YA
SIG(1) = ARSIN(KCAL * ((X2 - Xl) / B)) - SIGCAL
KCAL * (Xl - XCAL) - XA
X1CAL
X1CAL + A * COS(SIG(I )) + (B / 2) * SIN(SIG())
=
XG(I)
* (Y1 - YCAL) - YA
KCAL
=
YICAL
YG(I) = YlCAL + A * SIN(SIG(I )) - (B / 2) * COS(SIG(I))
Z(I) = I
YO(I) = 0
-
XGCAL
-
YGCAL
J = I
WRITE(6,11) J, H(I), SIG(I),
11 FORMAT ('0', 13, 4F10.4)
12 CONTINUE
C
C
C
PLOT
REDUCED DATA
(OPTIONAL, IP = 1
13, 13, 15
IF (NOP)
13 CALL NEWPLT ('M7754','8427'
IP = 0
IF (IP - NOP) 15, 15, 14
14 CALL PICTUR
XG( I),
FOR PLOT)
I
,'WHITE
,
'BLACK')
(15.0,10.0,'FRAME',5,'AMP',3,Z,H,IP,C.15,-6,Z,SIG,M,
$0.15,?-5, Z, XG, M,0.*15, - 3,Z ,YG,
M, 0.15,- 1,ZIYO,9M,
15 CONTINUE
C
C
C
YG(I)
CALCULATE WAVELENGTH
WRITE(6,16)
16 FORMAT ('1')
XLAM = G / (2 * PI * FREQ ** 2)
XLAM1 = XLAM
17 CORR = TANH((2 * PI * DEP) / XLAM)
ERR = XLAM / CORR - XLAMI
XLAM = XLAM - (0.7 * CORR * ERR)
WRITE(6,18) XLAM
18 FORMAT ('OXLAM = ',F9.5)
0 vKS)
- 0.0001) 19, 19,
IF (ABS(ERR)
19 CONTINUE
WRITE(6,20) XLAM, XLAM1, ERR
20 FORMAT ('OLAMDA = ',F9.5,'
$R = ',E9.3)
17
LAMDAI
=
ER
',F9.5,'
C
C
CALCULATE NUMBER OF DATA POINTS IN
1 CYCLE
C
CALL NPCINT (H,NI)
NR = NI
NP = 2 * NR + 1
WRITE(6,21) NP
21 FORMAT ('INPOINT =',14)
C
C
CALCULATE FIRST AND SECOND ORDER FOURIER COFFICIENTS
C
WRI TE(6,22)
ACOF(2)
22 FORMAT ('0 ACOF(1)
eCOF(3)')
$
IT = 3
CALL FORIER (HNR,IT,ACOFHBCOFHNOP)
CALL FORIER (XGNR,IT,ACCFXGBCOFXG,NOP)
CALL FORIER
(YGNRIT,ACOFYGBCOFYGNOP)
C
C
00
BCOF(?)
ACOF(3)
CALCULATE AND
PRINT KA,
MAGNITUCE
X/A,
PHASE
X/A
C
K = 2 * PT / XLAM
KA = K * D
**2)
2 + BCOFXG(2)
**
X = SQRT(ACOFXG(2)
2)
**
BCOFH(2)
+
2
**
HA = SQRT(ACOFH(2)
HA
/
X
XDH =
XGPHAS = ATAN2(BCOFXG(2), ACOFXG(2))
ACOFH(2))
HPHAS = ATAN2(BCOFH(2),
XA / XLAM - XGPFAS
PI
*
2
+
HPHAS
=
PHASE
WRITE(6,23)
PHASE
MAGNITUDE X/A
KA
23 FORMAT ('1
X/A
MAG
X
M
$AG A
PHASE X
PHASE A')
WRITE(6,24) KA,
XDH, PHASE,
X,
HA,
24 FORMAT ('O',F8.4,2F15.4,4F12.4)
25 CONT INUE
IF (NOP) 26,
26 CALL ENDPLT
27 CONTINUE
STOP
END
26,
27
XGPHAS,
HPHAS
SUBROUTINE FORIER (FNNR,!TACOFBCOFNOP)
A COF( 5), BCOF( 5), z(300),
HFCR2(300),
DIMENSION HFOR1(300),
$YO(3CC),
FN(300)
PI = 3.14159
NP = 2 * NR + 1
NX = 0
W = 2 * PI / NP
CALL FORIT (FN,NRIT,ACOFBCOF,IER)
IF (IER - 1) 3, 1, 1
IER
1 WRITE(6,2)
2 FORMAT (O0ER = ',11)
ACOF(3 ), BCOF(2) i BCOF (3)
3 WRITE(6,4) ACOF(1),
ACOF(2),
4 FORMAT ('0',F8.5,4F15.5)
C
C
PLOT
EATA WITH
1ST AND
IST + 2ND
FOURIER
COMP.
CURVES
(CPTIONAL)
C
IP = I
IF ( IP 5 DO 6 1 =
HFOR1()
HFOR2(I)
$SIN(2
*
.0
NOP) 7, 7, 5
1, NP
= ACOF(l) + ACOF(2) * COS(W * NX) + BCOF(2) * SIN(W
NX) + RCOF(3) *
= HFOR1(I) + ACOF(3) * COS(2 * W
W * NX)
*
NX)
I
YC(I) = 0
Z(1)
=
6 NX = NX + 1
CALL PICTUR (10.0,10.0,'FRAME ,5,'AMP ' , 3,ZFN ,NP,0. 15 ,-3 ,Z ,HFOR1,
$NP,O.10,-2,Z,HFOR2,NP,O.10,-1 ,ZYO,NP,0.0,KS)
7 CONTINUE
RETU RN
END
SUBROUTINE NPOINT (H,NR)
DIMENSICN H(300)
IF (H(i))
1 IF (H(l))
2
3
4
5
6
7
8
10
11
12
8,
3,
1
3, 2
8,
+ I
1 =
GO TO 1
IA = I
IF (HI)) 5, 6, 6
1 = 1 + I
GO TO 4
IF (HI)) 15, 15, 7
I =1 + 1
GO TO 6
IF (H(I)) 9, 10, 10
1 = I +
GO TO 8
IA = I
13, 13, 12
IF H I))
I = I + 1
GO TO 11
13 IF (H(I)) 14, 15, 15
14 1 = I + 1
GO TO 13
15 IB = I
IC = IB - IA
ID = IC / 2
IE = 2 * 1 0
IF (IC [E) 16, 16, 23
+- 1
+B
16 IG =
IF (H(18) - H(IA)) 17, 17, 20
19
17 IF (H(IG) - H(IB)) 18, 18,
1 NP = IC GO TO 24
19 NP = IC + 1
GO TO 24
20 IF (H(IG) - H(IB))
21 NP = IC + 1
GO TO 24
22 NP = IC - I
GO TO 24
23 NP = IC
24 NR = (NP - 1) / 2
RETURN
END
21,
21,
22
1
- 43 APPENDIX B
The theoretical values for the magnitude and phase of
were computed as a function of Ka using a Fortran program.
/A
From
Eq. 53, the relation for the magnitude is given by
x
A
S
(Q
2
+ R2 )
1/2
where Q, R, and S are as defined in Eq. 52, 53, and 51.
Eq.
55 the phase angle,
S = 0 + tan~
(B-1)
Also from
C is seen to be
(-R/Q)
(B-2)
44.
where e is defined by Eq.
A listing of the actual program used follows in which these
symbols were used
AD
boom draft
AMASS
added mass of fluid
BL
boom length
BM
boom mass
CHI
magnitude of X/A
DEP
water depth
PHASE
phase of X/A
TN
tension force
XKA
Ka
XL
length of tension members
-
44 -
YPD
pd, drag coefficient
YPH
tan 1 (G/-H)
YPM
PM,
added mass coefficient
I
DIMENSION XKA(50),
YPD(50 ), YPM(50) , YPH(50), XKA1(50),
XKA2(50),
$XKA3(50), XKA4( 50),
CHIl( 50)-, CH12( 50), CHI3(50),
CHI4(50),
$PHASE1(50) , PHASE2( 50), P HASE3( 50), PHASE4(50),
X(50),
Y(50),
$Z(50)
C
C
READ
IN
VALUES OF PD,
PM, ARCTAN(G/-H)
C
1
DO 2 I = 1, 29
READ (5,1) XKA(I),
FORMAT (4F10.4)
YPD(I),
2 CONTINUE
READ (5,3) Ti, T2, T3,
YPM(I),
YPH(I)
T4
3 FORMAT (4F10.2)
WRITE (6,6) Ti
WRITE (6,7)
C
C
CALL
SUBROUTINE
TO CALCULATE
MAGNITUDE
AND PHASE OF X/A
C
CALL CALC(T1,XKA, XKA1,YPOYPM,YPHCHIL,PHASEl)
WRITE (6,6) T2
WRITE (6,7)
CALL CALC ( T2 ,XKA, XKA2,YPD,YPMYPH,CHI2, P-ASE2)
WRITE (6,6) T3
WRITE (6,7)
CALL CALC(T3,XKA, XKA3,YPDYPMYPHCHI3,PHASE3)
WRITE (6,6) T4
WRITE (6,7)
CALL CALC(T4,XKA, XKA4,YPDYPM,YPHCHI4,PHASE4)
DO 5 1 = 1, 22
READ (5,4) X(I), Y(I), Zil)
4 FORMAT( 3F10.2)
5 CCNTI NUE
C
C
PLOT
CATA
C
CALL NEWPLT
(IM7754','8427','WHITE
',BLACK')
Ln
CALL PICTUR(15.5,10.0, 'KA',2,'MAGNITUDE X/A ', 13, X, Y, 22,O.2,-4, XKAl
$ ,CHI 1 ,29,).0 ,J,XKA2 ,CHI2 ,29,0. , J,XKA3, CHI3, 29,0.0,J ,XKA4,C HI4,
$29, 0 .0, J )
CALL PICTUR(15.5,10.I,'KA',2, 'PHASE X/A',9,X,Z,22,0.2,-4,XKAI,PHAS
$El,29,O. ,J, XKA2,PHASE2,29,0.O,J,XKA3,PHASE3,29,O.0,JXKA4,PHASE4,
$29,0.0,J)
CALL ENDPLT
6 FORMAT (1T
7 FORMAT ('0
$
STOP
END
=
KA
',F10.2)
MAG X/A
R
PHASE
S
OMEGA
LAMDA
TI-ETA')
ZETA
SUBROUTINE CALC(TXKAXKAN,YPOYPMYPHCH[,PHASE)
YPM(50), YPH(50),
YPD(50),
XKAN(50),
DIMENSION XKA(50),
$PHASE(50)
C
C
C
INITIAL
CHI (50),
SETUP
RHO = 1.0
G
980.35
9L = 30.0
BMASS = 39.8
PI = 3.14159
AD = 2.4
DEP = 15.0
XL = 50.0
AMASS = RHO *
PI * AD **
2 *
BL
/
2
I
C
C
CALCULATE VALUES
FOR MAGNITUDE
AND PHASE OF
X/A
C
DO 2 1 = 1,
29
XKAN(I) = XKA(I)
S = RHO * G * AD* BL * SQRT(PI * YPD(I) / 2)
XLAM = 2 * PI * AD / XKA(I)
I DEP) /XLAM))
SQRT(((G * 2 * PI) / XLAM) * TANH((2 * P
OMEGA
*
T / XL)
(2
2)
**
*
CMEGA
AMASS)
*
YPM(I)
Q= ((BMASS +
2)
**
OMEGA
* AMASS *
R = - (YPD(I)
CHI(I) = S / S QRT(Q ** 2 + R ** 2)
R / Q)
ZETA = ATANf2) - YPH(I)
THETA = (PI /
PHASE(I) = THETA + ZETA - PI
ZETA,
XLAM, CPEG A, Q, R, S,
WRITF (6,1) XKA(I), CHI(I), PHASE(I),
$THET A
1 FORMAT ('0' ,F5.2,4F10.4,3F15.4,2F8.4)
2 CONTINUE
RETURN
END
- 48 REFERENCES
1.
F. Ursell, "On the Waves Due to the Rolling of a Ship,"
Quarterly Journal of Mechanics and Applied Mathematics, Vol. 1,
1948, pp. 246-252.
2.
G. H. Watson, Theory of Bessel Functions, University Press,
second edition, 1952, pp. 77, 78, 329.
3.
J. Kotik, "Damping and Inertia Coefficients for a Rolling or
Swaying Vertical Strip," Journal of Ship Research, Vol. 7,
No. 2, October 1963, pp. 19-23.
4.
M. D. Haskind, "The Exciting Forces and Wetting of Ships in
Waves," Izvestia Akademii Nauk S.S.S.R., Otdelenie
Teckhnicheskikh Nauk, No. 7, 1957, pp. 65-79.
(English trans-
lation available as David Taylor Model Basin Translation No.
307, March 1962).
5.
J. N. Newman, "The Exciting Forces on Fixed Bodies in Wave,"
Journal of Ship Research, Vol. 6, December 1962, pp. 10-17.
6.
G. H. Keulegan and Lloyd H. Carpenter, "Forces on Cylinders
and Plates in an Oscillating Fluid," Journal of Research of
the National Bureau of Standards, Vol. 60, No. 5, May, 1958,
pp. 423-440.
pp
- 49 -tan 1(G/-H)
Ka
Table 1:
0.05
0.0043
1.0436
0.0050
0.1
0.0176
1.1041
0.0160
0.15
0.0423
1.1710
0.0345
0.2
0.0808
1.2334
0.0660
0.25
0.1350
1.2980
0.1075
0.3
0.2133
1.3690
0.1547
0.4
0.4302
1.4706
0.2854
0.5
0.7244
1.4750
0.4549
0.6
0.0046
1.3361
0.6476
0.7
1.1934
1.0837
0.8385
0.8
1.2505
0.8063
1.0060
0.9
1.2095
0.5722
1.1407
1.0
1.1211
0.4012
1.2439
1.1
1.028
0.2940
1.3216
1.2
0.940
0.2079
1.3796
1.3
0.852
0.1580
1.4232
1.4
0.778
0.1260
1.4560
1.5
0.707
0.1058
1.4811
1.6
0.641
0.0936
1.5003
1.7
0.585
0.0870
1.5151
1.8
0.534
0.0835
1.5267
1.9
0.489
0.0826
1.5357
2.0
0.451
0.0839
1.5428
2.5
0.330
0.1026
1.5615
3.0
0.245
0.1279
1.5676
3.5
0.195
0.1527
1.5684
4.0
0.158
0.1758
1.5692
4.5
0.127
0.1950
1.5700
5.0
0.099
0.2142
1.5707
Values of P D' PM, and -tan 1(G/-H) for Ka = 0.05
through 5.0.
Ka
Magnitude of X/A
0.05
0.10
0.15
0.20
0.725
5.077
2.979
1.569
-1.577
-1.634
-1.521
-1.542
0.25
1.209
-1.551
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
-1.550
-1.556
-1.560
3.00
3.50
4.00
4.50
1.056
0.912
0.846
0.800
0.767
0.740
0.718
0.700
0.678
0.657
0.641
0.624
0.610
0.599
0.586
0.575
0.563
0.549
0.467
0.380
0.300
0.234
0.183
5.00
0.141
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.50
Table 2
R
Phase of X/A
-1.933
-2.135
-2.352
-2.520
-2.655
-2.765
-8003.6
-2309.2
-6086.2
15959.2
26672.5
38096.7
60763.7
79552.3
88529.1
85457.8
74516.8
61727.9
50739.8
43739.8
37169.8
33908.3
32318.7
31981.9
32616.4
34036.0
35927.8
38289.8
41119.7
59082.9
81314.9
106158.0
132996.8
160451.0
-2.860
190033.9
-1.572
-1.584
-1.598
-1.614
-1.632
-1.657
-1.669
-1.693
-1.717
-1.747
-1.780
-1.815
-1.852
-1.892
Theoretical values for magnitude and phase of
for Ka = 0.05 through 2.0 with T = 0.142.
-7.2
-108.2
-516.4
-1519.9
-3427.0
-6768.8
-18823.9
-40004.0
-66756.8
-92592.7
-110908.7
-120689.1
-124300.4
-125376.4
-125066.2
-122804.5
-120764.4
-117582.4
-113712.8
-110264.6
-106572.6
-103013.6
-100008.8
-91471.5
-81492.8
-75671.9
-70072.7
-63364.8
-54882.9
S
5801.1
11736.2
18194.6
25146.6
32504.2
40857.2
58024.0
75294.2
88668.4
96641.9
98926.9
97291.6
93668.8
89695.2
85770.2
81656.8
78030.1
74384.4
70827.4
67662.3
64646.3
61862.5
59410.2
50819.4
43788.1
39065.2
35164.3
31526.4
27834.9
/A along with values for Q, R, and S
U,
c>
- 51 -
Run
Ka
Magnitude X/A
Phase X/A
A2
0.142
3.169
-1.329
A6
0.139
2.172
1.365
A10
0.144
1.885
-1.318
B2
0.202
1.770
-1.772
B6
0.202
1.425
-1.547
C2
0.291
0.963
-1.731
C5
0.299
1.027
-1.419
C8
0.300
0.910
-1.497
D2
0.467
0.836
-1.605
D3
0.536
0.775
-1.582
D4
0.536
1.043
-1.284
El
0.784
0.808
-1.433
E2
0.767
0.646
-1.461
F1
1.376
0.824
-1.004
G2
0.391
1.042
-1.349
G4
0.388
1.015
1.395
Table 3:
Experimental values of magnitude and phase of X/A
along with values of wavelength and wave height.
BOOM MODEL
T
Fc
2
x
T
L
WAVE
TRAIN
F=T sin a
Fig. 1
~
Schematic of constraint forces on model boom.
T
TENSION
MEMBERS
-53-
F2T
Ax
T
Fc n
FC
T sin an
=
T
T
--
(
a2
ax
I
Ax
Fig.
2
Schematic of constraint forces on full scale boom.
I
Fig. 3
Close-up photograph of boom in wave tank
during experimental run C5 (Ka = 0.30).
UPI
AJ
Fig. 4
.
Overall view of boom in wave tank showing elastic constraint support arms and
capacitance wave height guage (on carriage behind the boom model).
a
b
0i
c
Fig. 5
d
run C5 (Ka = 0.30). Photographs were
experimental
during
Sequence of boom positions
the same view as the data films.
showing
camera
taken with a motor-driven 35 mm still
-$7-
11 YI2
XCAL 2
XCAL
I
YCAL I
d
I
X2
I
Fig. 6
t
Drawing of measurement grid lines and boom pointer
illustrating measurement points.
"M I---E
am-
5.5
5.0
= 0.0
r = 0.071
T
4.5 -I
2
3 r = 0.142
4 r = 0.281
4.03.53.0:>
2.50
4
2.0-
3
2
0.5 --
0.01
0.0
Fig. 7
0.5
1.0
1.5
2.0
2.5
Ka
Plot of the theoretical values of the magnitude of
T = 0, 0.071, 0.142, 0.281.
3.0
3.5
4.0
4.5
X/A for Ka = 0.05 through 5.0 with
5.0
-1.20
I r
0.0
2 r = 0.071
3 r = 0.142
4 T = 0.281
-1.40-7r/2
-1.60
4
3
2
-1.80 -
0
X
-r
-2.00 -
-2.20 I
r
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Ka
Fig. 8
Plot of the theoretical values for the phase of x/A for Ka = 0.05 through 5.0 with
T = 0, 0.071, 0.142, 0.281.
I.
- -
4
-0
ft 10 , -,
-
I. 8
1.6
1.4
+
+
-4
1.2
RUN C5
+ DATA
I St FOURIER COMPONENT FIT
I st + 2 nd FOURIER COMPONENT FIT
_++
1.0
0.8
E
U
0.6
w
-1
/+
0.4
0
I-J
0~
I.I
0.2
+
0.0
-0.2
-0.4-
a\
-0.6-
0
w -0.8I
Lii
-
1.0-
-
1.2
-
1.4_
-
1.6
-
1.8-
++-,
_
. *01
-2.0
-2
-29
0
Fig. 9
2I
I I
4I
I
6I
I| |I
8I
10
I
12
I
14
I
20 22
18
16
FRAME NUMBER
24
26
28
30
32
34
36
38
40
Example of wave height data from run C5 with curves using the first fourier coefficients
and second fourier coefficients.
and the first
___________________
I-
2.0
RUN C5
1.8-
+ DATA
st FOURIER COMPONENT FIT
Ist + 2nd FOURIER COMPONENT FIT
1 .2 I .0I
E
0.8 -
0.6-
/
0.4
4-
0.2 -
a-
0.0
/+
-1%
-0.8
-
1.0
I
--
- 1.2 - 0.4-
- 1.6 0
2
4
6
8
10
12
14
16
18
FRAME
Fig. 10
20
22
24
26
28
30
32
34
36
NUMBER
Example of surge displacement data from run C5 with curves using the first fourier
coefficients and the first and second fourier coefficients.
38
40
~i
it'IiI~fl'1JuTLJI]I
L
I 111J11in111U1i1h111L5E1.
5.5
5.0
4.5
4.0
3.5 |
+A
LU
3
.0
- 2.5
z
0
A6+
< 2.0 }
-IO
AIO
B2
1.5
B6
1.0
C 2+++*-
Fig. 11
+D4
F
C8 G4 D 2
0.50.0
0.0
C G2
I
0.1
I
0.2
I
0.3
I
I
0.4 0.5
++E
D
I
0.6
I
0.7
I
I
0.8 0.9
+ FI
I
1.0
Ka
I
1.1
I
1.2
I
1.3
I
1.4
I
1.5
I
1.6
I
1.7
I
1.8
I
1.9 2.0
Plot of the theoretical values of the magnitude of X/A with the experimental data points
for Ka = 0.05 thru 2.0 and T = 0.142.
0.0
-0.20-
0.40
- 0.60L
-
0.80
C')
0
-1.00-
~0
0
-
1.20
- 1.40-
1.6
-
1.80
+FI
A2*AIO
+A6
Lii
(1~)
I
0~
+
G2
+C5 tG4
+D4
+C8
+
+EI
E2
D2
+C2
I
B2
- 2.00
2.202.402.602.803.00 0.0
Fig. 12
I
I
I
I
I
I
I
I
I
I
I
Q1
0.2
Q3
04
0.5
0.6
0.7
0.8
0.9
1.0
Ka'
1.1
I
III
1.2
1.3
1.4
I
1.5
I
1.6
I
1.7
1.8
1.9
Plot of the theoretical values of the phase of X/A with the experimental data points
obtained for Ka = 0.05 thru 2.0 and T = 0.142.
2.0